Fundamentals of direct limit Lie theory

We show that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces. This enables us to push all basic constructions of finite-dimensional Lie theory to the case of direct limit groups. In particular, we obtain an analogue of Lie's third theorem: Every countable-dimensional real or complex locally finite Lie algebra is enlargible, i.e., it is the Lie algebra of some regular Lie group (a suitable direct limit group).